1 \chapter{Hydrodynamic effects in fast AFM single-molecule force measurements}
5 {\Large M\"uller notes} \\
8 We had some trouble with the notation in \citet{janovjak05}, so I'll
9 try and clear some things up\ldots
12 \begin{tabular}{r|l|l}
13 M\"uller \Bstrut & Trevor & Meaning \\
15 $z_{surface}$ \Tstrut & $z_{surface}$ & Distance from the surface to the equilibrium cantilever position (increases on pulling) \\
16 $z_{cantilever}$ & $z_{cantilever}$ & Cantilever deflection from its equilibrium position (downward deflection positive) \\
17 $h$ & $h$ & $h = z_{surface} - z_{cantilever}$ the distance between the tip and surface \\
18 $v_{tip}$ & $v_{tip,surface}$ & $v_{tip,surface} = dh/dt$, tip velocity relative to the surface \\
19 & $v_{eq,surface}$ & $v_{eq,surface} = dz_{surface}/dt$, pulling speed \\
20 & $v_{tip,eq}$ & $v_{tip,eq} = dz_{cantilever}/dt$, tip velocity relative to its equilibrium position \\
21 $F_{measured}$ & $F_{measured}$ & Measured force deflecting the cantilever \\
22 $F_{net}$ & $F_{protein}$ & Force applied to stretch the protein \\
23 $F_d$ & $F_d$ & Drag force acting on the cantilever \\
24 $\Delta F$ & $F_{d:tip,eq}$ & Drag force due to only to $v_{tip,eq}$ \\
25 & $F_{d:eq,surface}$ & Drag force due to only to $v_{eq,surface}$, the drag on an untethered cantilever \\
26 & $F_{meas,zeroed}$ & $ F_{meas,zeroed} = F_{measured} - F_{d:eq,surface}$, defined for zero force in the detached region
32 F_d &= \frac{6 \pi \eta a_{eff}^2}{h + d_{eff}} \cdot v_{tip} \label{mul_Fd} \\
33 h &= z_{surface} - z_{cantilever} \\
34 v_{tip} &= \frac{dh}{dt} \\
35 \Delta F &= F_d(v,h) - F_d(v_{tip}, h) \label{mul_delF} \\
36 F_{net} &= F_{measured} + \Delta F \label{mul_Fnet}
39 Trevor derivations: \\
40 For \cref{mul_delF}, we assume that all the fluid in the cell moves with the surface
41 (i.e., fluid flow does not depend on height above the surface).
42 So the drag force is proportional to the speed of the tip relative to the surface.
44 F_d = D(h) v_{tip,surface}
46 Where $D(h)$ is some constant that can depend on $h$ (like $6 \pi \eta a_{eff}^2 / (h + d_{eff})$).
47 This is \xref{janovjak05}{equation}{mul\_Fd??}.
48 Substituting in $v_{tip,surface} = v_{eq,surface} - v_{tip,eq}$ we have
50 F_d &= D(h) v_{eq,surface} - D(h)v_{tip,eq} = F_{d:eq,surface} - F_{d:tip,eq} \\
51 F_{d:tip, eq} &= F_{d:eq,surface} - F_d
53 This is \xref{janovjak05}{equation}{mul\_delF??}.
54 The measured force deflecting the cantilever is then
56 F_{measured} &= F_{protein} + F_d \\
57 F_{protein} &= F_{measured} - F_d = F_{measured} - (F_{d:eq,surface} - F_{d:tip,eq}) \\
58 &= F_{meas,zeroed}' + F_{d:tip,eq} = F_{meas,zeroed}' + D(h)v_{tip,eq}
60 This is \xref{janovjak05}{equation}{mul\_Fnet??}.
62 The treatment assumes the drag force on a detached cantilever doesn't depend on distance (see dashed line in \xref{janovjak05}{figure}{4b,c}), which doesn't make sense because
64 F_{d:eq,surface} = D(h)v_{eq,surface}
66 And $D(h)$ depends on $h$. Therefore, this treatment uses $F_{meas,zeroed}'$, not $F_{meas,zeroed}$, where
68 F_{meas,zeroed}' = F_{measured} - F_{d:eq,surface,h\approx300nm}
69 = F_{meas,zeroed} + [D(h) - D(300nm)]v_{eq,surface}
71 What can we do about this?
73 The correction from $F_{meas,zeroed}'$ (solid line in \xref{janovjak05}{figure}{3a}) to $F_{protein}$ (dashed line) comes from adding $F_{d:tip,eq}$, which is why $F_{protein} = F_{meas,zeroed}'$ when
75 0 = F_{d:tip,eq} \propto v_{tip,eq} = \frac{dz_{cantilever}}{dt} \propto \frac{dF_{meas,zeroed}}{dt}, \\
77 why $F_{protein} < F_{meas,zeroed}'$ when the cantilever is rebounding ($v_{tip,eq} < 0$), and
78 why $F_{protein} > F_{meas,zeroed}'$ when the cantilever is loading the protein ($v_{tip,eq} > 0$).
80 This last ($F_{protein} > F_{meas,zeroed}'$ on loading) is why the raw
81 measurement \emph{underestimates} the unfolding force.